Find potential difference `V_(AB)` between `A(0,0,0) and `B(1m,1m,1m)` in an electric field a. `E-yhati+xhatj` b. `E=3x^2yhati+x^3hatj`

Answer

A, B, C

Explanation

a. `dV=-E.dr`
`:. int_(B)^(A) dV=-int_(((1,1,1)))^(((0,0,0)))(yhati+xhatj).(dxhati+dyhatj+dzhatk)`
or `V_A-V_B=-int_(((1,1,1)))^(((0,0,0))) (y dx+x dy)`
or `V_(AB)=-int_(((1,1,1)))^(((0,0,0)))d(xy)` [as `y dx+x dy=d(xy)`]
`:. V_(AB)=[xy]_(((1,1,1)))^(((0,0,0)))=1V`
b. `dV=-E.dr`
`:. int_B^A dV=-int_(((1,1,1)))^(((0,0,0)))(3x^2yhati+x^3hatj)*(dx^hati+dyhatj+dzhatk)`
or `V_A-V_B=-int_(((1,1,1)))^(((0,0,0)))(3x^2ydx+x^3dy)`
`=-int_(((1,1,1)))^(((0,0,0)))d(x^3y)`
`:. V_(AB)=-[x^3y]_(((1,1,1)))^(((0,0,0)))=1V`